Logarithmic Functions

Terms

Problems

Logarithmic Functions

Like many types of functions, the exponential
function has an inverse. This inverse is
called the logarithmic function.

logax = y
means
ay = x
.

where
a
is called the base;
a > 0
and
a≠1
. For example,
log232 = 5
because
25 = 32
.
log5 = - 3
because
5-3 =
.

To evaluate a logarithmic function, determine what exponent the base must be
taken to in order to yield the number
x
. Sometimes the exponent will not be a
whole number. If this is the case, consult a logarithm table or use a
calculator.

Since no positive base to any power is equal to a negative number, we cannot take the
log
of a negative number.

The graph of
f (x) = log2x
looks like:

Figure %:
f (x) = log2x

The graph of
f (x) = log2x
has a vertical asymptote at
x = 0
and passes
through the point
(1, 0)
.

Note that
f (x) = log2x
is the inverse of
g(x) = 2x
.
f
o
g(x) = log22x = x
and
g
o
f (x) = 2log2x = x
(we will learn why this is
true in Log properties). We
can also see that
f (x) = log2x
is the inverse of
g(x) = 2x
because
f (x)
is the reflection of
g(x)
over the line
y = x
:

In general,
f (x) = c·loga(x - h) + k
has a vertical asymptote at
x = h
and passes through the point
(h + 1, k)
. The domain of
f (x)
is and the range of
f (x)
is . Note that this domain and
range are the opposite of the domain and range of
g(x) = c·ax-h + k
given in Exponential Functions.